Recent outbreaks have highlighted the need for new tools and appropriate guidance to protect these workers and their families. Unlike chemical hazards, the health risks associated with specific exposure concentrations have not been well characterized for most infectious agents. Furthermore, methods to quantify workers’ risk of infection based on job tasks and activities are not well developed.
To maximize the effectiveness of engineering controls, administrative controls, and personal protective equipment in reducing exposure to pathogens in the work environment, industrial hygienists must first understand how infectious diseases are transmitted, how infectious particles move through the environment, and what types of interactions allow infections to spread between workers and the community. This is especially important in situations where resources are limited and quick decisions are necessary. The use of mathematical models to assess workers’ risk of exposure is a cost-effective approach to protecting workers from infectious disease.

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WHY USE MODELS FOR INFECTIOUS DISEASES?

Mathematical models have been used for decades to predict the spread of infectious diseases and to develop strategic interventions to control epidemics. Today, many free, online modeling tools are available to examine how diseases spread through populations (see Table 1). These models, however, typically emphasize community outbreaks and epidemics. Few models are available that apply to the occupational setting. Ideally, industrial hygienists should be able to quantify, rapidly and inexpensively, the number of viable infectious particles that a worker is exposed to through various exposure routes. Unfortunately, however, bacteria, viruses, and other infectious agents cannot be easily or quickly detected and measured.
Even when measurements are feasible, it is often difficult to determine what fraction of infectious particles are viable and able to cause disease. Through modeling, industrial hygienists can make data-driven exposure estimates that capture spatial and temporal variability, decide which risk management measures to adopt and which locations within the work environment are of concern, and prioritize groups of workers for interventions. Models can also predict future exposures and outcomes, enabling the hygienist to be proactive when considering strategies to protect workers. Table 1. Online Tools for Infectious Disease Modeling

*Tap on the table to open a larger version in your browser.*

Although mathematical modeling for infectious disease in occupational settings is a nascent field, some models can be used to describe and predict workplace infections. Examples include models that identify risk factors for occupational tuberculosis transmission to sheriffs during a prison outbreak, risks of various infectious diseases to healthcare workers in hospitals, and risk of Legionnaires’ disease to workers in the spa industry. These types of models are particularly useful because they can be used to determine which parameters have the potential to impact transmission the most. Unfortunately, it is not always easy to generalize these models and apply them to other scenarios.
Aside from the obvious impacts to workers, occupational exposures to infectious diseases have important implications for communities. In a hospital setting, healthcare workers have frequent contact with patients, so efforts to decrease worker infection also decrease patient and family exposure to hazardous microbial agents. The same is true for other occupations. For example, livestock-associated strains of methicillin-resistant

Staphylococcus aureus

(MRSA) have been found in the general population, even among those who have no links to livestock, which suggests that farm workers may transmit infections acquired in the workplace to the community.
MAKING MATHEMATICAL MODELS MORE ACCESSIBLE

While modeling can be useful in describing and predicting infectious disease transmission, development of an operational mathematical model doesn’t guarantee it will be used successfully. Many practitioners are suspicious of models. Potential users may not understand how a model works or how its outputs can be used for decision making. Furthermore, they may be unsure of the model’s accuracy and wary of its assumptions. To overcome this perception of models as mysterious, models for disease transmission must be validated, transparent, accessible, and understandable.
Models should be simple and easy to access.

Many mathematical models for disease transmission are presented in a way that is inaccessible to non-academics. Papers describing mathematical models of infectious disease risk often present long equations that can be calculated using only expensive, specialized statistical software, making it difficult for practitioners to apply them for decision making. Free, computer-based modeling tools for infectious disease transmission are available, but these typically focus on global, national, or large-scale outbreaks and have limited utility for identifying risks to individuals or work groups. Easy-to-use and easily accessible tools are needed for use in workplace settings.
Models should be useful for making decisions.

For a model to be useful, it must be able to guide the user’s decision making. Good models allow users to explore how altering certain variables will result in changes in worker risk. The following are examples of questions that models can help answer:
- What is the effect of increasing general ventilation rates by 30 air changes per hour?
- Will decreasing the time workers spend in patient rooms by 30 percent significantly lower worker infection rates?
- Which regions within an airplane cabin present the highest risk of exposure to disease given the seat assignment of an infected traveler?
- During which activities of industrial farming are workers at highest risk of contracting zoonotic diseases?
- What type of glove is needed to reduce exposure risk to some
*de minimis*level? - Are full-body suits the most efficient and appropriate control measure for protecting workers responding to an Ebola outbreak?

Well-designed sensitivity analyses can identify which variables are most influential in determining worker risk. Furthermore, model outputs can be linked to control strategies to aid in decision making.

Models that are scalable can improve implementation.

Another reason mathematical models may not be fully utilized is that most are not set to the scale needed for decision makers to utilize them. Risk management happens at many levels; models for use during an emergency response, for example, should be practical for individual responders, incident commanders, and city, state, or county officials. A multi-level approach that describes the movement of infectious particles within a small space (for example, a hospital ward or aircraft carrier), the spread of infection within a workplace among similar exposure groups, and the spread of disease on a larger scale and from the workplace into the community would lend itself well to decision makers at all levels. Guidance is needed to help practitioners understand what types of models can be used in which situations. Figure 1 demonstrates various levels in which modeling can be applied. Models can be scaled to predict the risk to individuals, groups of workers, and populations.Figure 1. Scalable models can predict risks to individuals, workers, and populations.

*Tap on the figure to open a larger version in your browser.*

INFECTIOUS DISEASE MODELS IN PRACTICE

SEIR Model

The Susceptible-Exposed-Infectious-Recovered model (Figure 2) is one of the most commonly used models to describe the spread of infectious diseases. The model uses a set of differential equations to predict the proportion of the population who are susceptible, exposed, infectious, and recovered/immune to a disease at a given time: In Figure 2, the rates of movement between compartments are defined by the variables λ, f, and r. This type of modeling is often used to predict the peak of infection, determine a target for vaccination programs, and determine the basic reproduction number, Ro (new infections generated by a primary infected individual in a completely susceptible population). Many variations of this model exist, and not all compartments need to be present. For example, some diseases are better described by an SIS model, where susceptible individuals are assumed to immediately become infectious; once the infection has resolved they are assumed to be completely susceptible again, rather than recovered/immune.

Figure 2. The SEIR model predicts the proportion of the population who are susceptible, exposed, infectious, and recovered/immune to a disease at a given time.

*Tap on the figure to open a larger version in your browser.*

**RESOURCES**

Applied Occupational and Environmental Hygiene

: “Occupational Tuberculosis among Deputy Sheriffs in Connecticut: A Risk Model of Transmission” (November 1999).
Journal of Occupational and Environmental Hygiene

: “Occupational Exposures to Influenza among Healthcare Workers in the United States” (March 2016).
Journal of Water and Health

: “Legionnaires’ Disease: Evaluation of a Quantitative Microbial Risk Assessment Model” (June 2008).
Oxford University Press: An Introduction to Infectious Disease Modellin

g (2012).
PLOS One

: “A Metapopulation Model to Assess the Capacity of Spread of Methicillin-Resistant

Staphylococcus Aureus

ST398 in Humans

” (October 2012).
*Markov Modeling*

A Markov chain is a simple way to model a stochastic process. In a Markov process, entities can move between a defined number of states. The rate at which an entity moves from one state to any other state is defined by a transition probability. At each time step, the model applies the transition probability to determine what occurs at the next time step. In this way, a Markov model can predict events that occur randomly over time. For infectious diseases, a Markov model can theoretically be scaled to predict the transport of infectious particles within a workplace through three-dimensional space using principles of aerosol physics and airflow to estimate the transition probabilities. Knowledge gained from chemical transport models for inhalation exposure models can be embedded into Markov models for infectious diseases. Markov techniques can also be applied to model the disease status (that is, susceptible, infectious, or recovered) of individuals interacting randomly with others in a population, or the probability of an infection moving from a workplace into the community and vice versa.

PREDICTING SPREAD OF DISEASE

Mathematical models for infectious diseases are not commonly used in the field of occupational health. However, such models have a tremendous potential to predict the spread of infectious diseases in the workplace and inform control decision-making. Modeling tools that are scalable, tied to control measures, and simple to use and understand would increase the use and effectiveness of this approach.

**MELISSA SEATON, MS,**is a PhD student in the Department of Environmental Health and Engineering at the Johns Hopkins School of Public Health. She can be reached viaemail

.

**GURUMURTHY RAMACHANDRAN**is a professor in the Johns Hopkins Department of Environmental Health and Engineering and director of the Johns Hopkins Educational and Research Center funded by NIOSH.Send feedback to

*The Synergist*. Using Mathematical Models to Guide Risk Management for Infectious Diseases in the Occupational Setting

Potential for

PREDICTION